A data communications system oftentimes employs a multidimensional signaling scheme. Examples of multidimensional signaling systems are shown in U.S. Pat. No. 4,084,137 issued Apr. 11, 1978 to G. R. Welti, which is hereby incorporated by reference. In such a system, each possible value of an input word, which includes a plurality of data bits, is assigned to a multidimensional signal point. Such a signal point is drawn from a signal constellation, whose desirability is in part measured by a peak-to-average-power ratio (PAR). Specifically, PAR is defined as the ratio of the maximum signal power to the average signal power, needed to transmit a signal point of the constellation. As is well-known, the susceptibility of the constellation to nonlinear distortions and perturbations incurred in a channel varies directly with the PAR. Thus, a constellation having a low PAR is desirable in that its signal points are more likely to be accurately recovered despite the presence of the nonlinear distortions and perturbations which adversely affect the signal point transmission.
The signal points of a signal constellation in a multidimensional signaling scheme are uniformly distributed within an enclosed domain in an N-dimensional space, where N.gtoreq.2. It is well-known that the shape of this enclosed domain affects the average signal power. Thus, it is desirable to shape the constellation in such a manner that the average signal power is minimized so as to conserve the transmission power. The term "shape gain" refers to the decrease in the average signal power by virtue of the shape of the constellation. It is specifically defined as the relative reduction in the average signal power inherent in a signal constellation having a certain shape with respect to a cube having the same volume and dimensionality N as that constellation. In fact, a spherical signal constellation provides the theoretically maximum or full shape gain for each dimensionality N. Furthermore, the shape gain of a constellation can also be improved by increasing the dimensionality of the constellation, as taught by C. E. Shannon in his classic paper "Communication in the Presence of Noise," Proc. IRE, Vol. 37, January, 1949, pp. 10-21 , which is hereby incorporated by reference. In fact, the shape gain of a spherical constellation approaches 1.53 dB as the dimensionality N becomes infinity. This shape gain is referred to as the "asymptotic shape gain."
In practice, the realization of the asymptotic shape gain is, nevertheless, thwarted by problems in realizing the circuitry for addressing the signal points of the signal constellation. For a given signaling rate, the number of required signal points grows exponentially with the dimensionality of the constellation. As a result, the complexity of the addressing circuitry increases tremendously and the speed thereof, accordingly, decreases. This being so, as the dimensionality becomes substantially large, such addressing circuitry invariably becomes too slow to support the given signaling rate.
Attempts have been made to solve the circuitry-complexity problem by simplifying the scheme for addressing the signal points. One such attempt for block coding uses an addressing technique as described in U.S. Pat. No. 4,507,648, issued Mar. 26, 1985 to Conway, et al., which is hereby incorporated by reference. Although this technique achieves a relatively simple addressing scheme, it unduly restricts a communications system to the use of a particular type of constellation which imposes an undesirably high PAR. Another attempt involves convolutional coding and uses a technique as proposed in U.S. Pat. No. 4,713,817, issued Dec. 15, 1987 to Wei, which is also hereby incorporated by reference. In accordance with this technique, an N-dimensional signal point is formed by concatenating a plurality of constituent 2-dimensional signal points in a prescribed manner, and each N-dimensional signal point is addressed in terms of these constituent 2-dimensional signal points. Significantly, the constellation resulting from using this technique provides a relatively low PAR, but, nevertheless, an undesirably small shape gain.
Accordingly, it is desirable to have a technique which not only provides a low PAR and a high shape gain, but can also efficiently address the signal points.